Basics Polyakov’s Quantum Surface Theory on the Formalism of Functional Integrals and Applications

Luiz C L Botelho

doi:10.5402/2012/674985

Abstract

We propose an alternative view to the covariant Polyakov’s string path integral. In our new approach we clarified the role of the Liouville model on string theory for Q.C.D. and Dual Models. In the appendices, we present additional material, difficult to find in the specialized path integral literature on the detailed evaluation of the (Q.C.D.) fermion determinant, the path integral proof of the Atiyah-Singer index theorem on Riemannian Manifolds, and the role of the expansion on Polyakov’s String path integral, all containing important new results, insights on theses topics.

Highlights

Since its inception seventy years ago, nonabelian gauge theories have been shown to be the most promising mathematical formalism for a realistic description of strong interactions and even formulated on its supersymmetric version; it became an attractive attempt for unify Physics
In strong interaction Physics the picture image of a mesonic quantum excitation is, for instance, a wave quantum mechanical functional assigned to a classical configuration of a space-time trajectory of a pair quark-antiquark bounding a space-time nonabelian gluon flux surface of all topological genera connecting both particle pairs: the famous t’HooftFeymman planar diagrams 1
It appears appealing for mathematical formulations to be considered directly as dynamical variables or wave functions in this Faraday line framework for nonabelian gauge theories, the famous quantum Wilson Loop, or quantum holonomy factor associated to ISRN High Energy Physics a given space-time Feynman quark-antiquark closed trajectory C in a SU Nc Yang-Mills quantum field theory

Summary

Introduction

Since its inception seventy years ago, nonabelian gauge theories have been shown to be the most promising mathematical formalism for a realistic description of strong interactions and even formulated on its supersymmetric version; it became an attractive attempt for unify Physics. × DF φ ξ which would lead to the same expression 3.28 but with the “reduced” anomaly conformal coefficient 26D to 25D, if one could disregard the infinite piece on 3.29 and if the “Fujikawalike” evaluation of the functional jacobian could be done more invariant At this point it appears that the above 2D-Liouville-Polyakov path integral only makes sense at the classical level which is formally equivalent to evaluate all the observables in the φ ξ -theory at the limit of D → −∞ 1. It is very important on applications to the Dual model theory for strong interactions off-shell Scattering Amplitudes to have a covariant regularized form for the formal Green function of the Beltrami-Laplace operator in the conformal gauge gab ξ eφ ξ δab.

Path-Integral Quantization of the Nambu-Goto Theory of Random Surfaces

Pseudoclassical Path Integrals

Field Model on the Botelho Sigma Model Parametrization

Ω2 classical term μ2 p0 2